That synopsis is wrong. T2 takes place in 1995, 2 years before the original Judgement Day which was indeed supposed to be in 1997 as you say (but was postponed due to the events of T2). John Connor is 10 years old in Terminator 2, and was born in 1985, 9 months after the events of Terminator.
It doesn't really matter, because all 3 movies, this show, and any other incarnations all introduced continuity errors. Trying to make sense of it doesn't work, because too many things directly contradict other things. For example, even though John is supposed to be 10 in Terminator 2, in Terminator 3 he gives his age as being thirteen during those events, which lines up with the 1997 date of Judgement day. But then, he later tells the Terminator that he thought they stopped Cyberdine "over ten years ago", which lines up with 1994 or earlier, not 1995.
On top of all that, T2 was released in 1991 and its events are supposed to take place in 1995. The original Terminator was supposed to be released in 1983, but filming was delayed a year and instead was released in 1984, as told by Cameron in the DVD commentary. A cop tells Reese at the beginning of that film that the date is Thursday May 12, 1984, when May 12 wasn't a Thursday in 1984 (May 12 was a Thursday in 1983, when the script was written).
I could go on, but the point is that its useless to argue over the timeline in the Terminator franchise.